There are a large number of groups who use the label “Catholic”… a certain sub-group of Orthodox do so (almost as provocateurs) who are not in union with Rome.
The so-called “Independent Catholic” movement, including at least 3 rites amongst them, use Catholic Liturgies but are not in union with the Pope, and often, not even with other IC groups.
The Eastern Orthodox informally use the term Orthodox Catholic, but far more often use an enthic identifier, such as Romanian Orthodox. The Eastern Orthodox Communion considers themselves the remnant of the Catholic Church, and the Romans to be either heretics and/or schismatics.
The Old Catholic movement is a schismatic synod; their genesis looks a lot like the SSPX’s… disagreement with a council of the church, followed by excommunication and removal thereof, followed by formal schism as they turned away from rome in disagreement.
The Uniates or Unia (tho those terms are disliked by many) are those Orthodox churches that came into union with Rome. The Orthodox generally consider them non-Orthodox, and they are fully Catholic, in unity with Rome. 22 named churches, covering 6 rites. Amongst them, the Byzantine Rite (Including the Byzantine Catholic Metropolitan Church of Pittsburgh, the Ukrainian Catholic Church,
FSSJ, SSPX: not fully in union, nor truly separated yet. They accept the pope as the pope, but disobey the pope anyway. Both want to return to the liturgy as of 1960 or so. The SSPX split from the Romans, and the FSSJ from the Ukrainians.
All the above claim to be catholic. Only the Unia and Rome are truly Catholic in the sense you ask.
The Melkites are Byzantine Rite, in union
The Maronites are Syrian Rite, and in Union
The Chaldeans are Chaldean Rite (or Syrian Rite, depending on which resource one looks in), and in union
Byzantine Catholic is used by both the Metropolitan Church of Pittsburg (in union), and the Byzantine Catholic Church, Inc., an independent catholic movement church (not in union).